Kullback-Leibler Divergence: Measuring the Gap Between Probability Distributions in Transfer Matrix Methods

Previous sections highlighted the limitations of traditional transfer matrix methods when dealing with sample-dependent noise and inaccurate posterior probability estimation. These limitations arise primarily from the disparity between probability distributions A and B. To address this, we aim to introduce a metric to quantify the discrepancy between these probability distributions.

For this purpose, we introduce Kullback-Leibler divergence (KL divergence), a metric that measures the difference between two probability distributions. KL divergence is an asymmetric measure, quantifying the information loss when transitioning from probability distribution B to probability distribution A.

Specifically, KL divergence is defined as the cross-entropy between probability distributions B and A minus the entropy of probability distribution B. The mathematical expression is as follows:

KL(A || B) = ∑ P(A) log(P(A)/P(B)) - ∑ P(A) log(P(A))

Here, P(A) and P(B) represent the probability values in probability distributions A and B, respectively.

By calculating KL divergence, we obtain a quantifiable index that measures the discrepancy between probability distributions A and B. A smaller KL divergence value indicates a smaller difference between the two probability distributions, while a larger value signifies a greater difference.

In this article, we will utilize KL divergence as an evaluation metric to gauge the impact of sample-dependent noise on transfer matrix methods. Our goal is to analyze the results of KL divergence to assess the accuracy of posterior probability estimation and subsequently enhance the effectiveness of transfer matrix methods.